Second quantization

The second quantization (often also called occupation number representation, in quantum field theory also field quantization ) is a method for the quantum mechanical treatment of many-particle problems , in particular the processes in which particles are created or destroyed. It was developed shortly after the discovery of quantum mechanics ( first quantization ) in order to be able to describe photons and their creation and destruction in a quantum mechanical way. The photons appear in the second quantization as the field quanta of the quantized electromagnetic field, which led to the third name given. When it was discovered in the 1930s that “material” particles can also be created and destroyed, the scope of the method was expanded to include all particles. In physics, the clear contrast between particles and waves in its earlier fundamental meaning was thus canceled.

The second quantization is used in the field of solid state physics , quantum field theory and other many- body theories . It is often the most appropriate framework for treating physical problems theoretically.

The method comes from Paul Dirac (1927).

Basic concepts

The following is a brief summary of some of the major new terms and their immediate consequences. As expected, the term “particle” denotes something other than the equivalent term in classical mechanics or everyday language. Here, like other apparently familiar terms - “identity”, “transition”, “annihilation” - it is used more metaphorically: The aim is to be able to use the mathematical definitions presented here in a catchy way when a quantum mechanical explanation is macroscopic observable phenomena are required, for example discoloration, the increase in the current intensity in a semiconductor or the directional distribution of radiation.

• The state of the system considered, as in the ordinary quantum mechanics by a normalized vector in a Hilbert space specified, constructed as a so- Fock space containing the states with different particle numbers.
• There is a state without any particle, the absolute vacuum , symbol . The vacuum state is normalized, so it must not be confused with the zero vector . ${\displaystyle \vert O\rangle }$ ${\displaystyle \langle O\vert O\rangle =1}$
• For each type of particle there is a creation operator that puts it into the world in a defined state, symbol (for a different type of particle, etc.). The 1-particle state with one particle in the state is then given by . The 2-particulate state with a second particle of the same kind, but in the state , is then added by again applying the generator: . For further particles further creation operators accordingly. ${\displaystyle a^{\dagger }}$ ${\displaystyle b^{\dagger }}$ ${\displaystyle p}$ ${\displaystyle a_{p}^{\dagger }\vert O\rangle }$ ${\displaystyle k}$ ${\displaystyle a_{k}^{\dagger }a_{p}^{\dagger }\vert O\rangle }$
• Since the “ ” -particles are identical to each other, no other state may result if they are exchanged in the order in which they were created. At most, the sign may change. That is guaranteed by the conditions ${\displaystyle a}$

${\displaystyle a_{k}^{\dagger }a_{p}^{\dagger }=+a_{p}^{\dagger }a_{k}^{\dagger }}$for bosons ("interchangeable")

${\displaystyle a_{k}^{\dagger }a_{p}^{\dagger }=-a_{p}^{\dagger }a_{k}^{\dagger }}$for fermions ("non-exchangeable").

Generators of different types of particles are always interchangeable. Two things are achieved early in the formalism:

1. The absolute indistinguishability of the same particles is built in. The particles are no longer even given a number in order to be able to distinguish their coordinates from one another.
2. Boson states are always symmetric against exchange, fermion states are always antisymmetric . The Pauli principle is automatically taken into account and the different quantum statistics inevitably result.

• The operator for the annihilation of a particle in the state is . An application example: Here the destruction can be an existing particle in a vacuum, the empty vacuum back . The annihilator is the operator hermitically adjoint to the creator . You can see that this is the right thing to do. B. when calculating the norm of , d. H. for the scalar product with its adjoint vector  : ${\displaystyle p}$ ${\displaystyle \,a_{p}}$ ${\displaystyle a_{p}\,a_{p}^{\dagger }\vert O\rangle =\vert O\rangle }$ ${\displaystyle a_{p}^{\dagger }\vert O\rangle }$ ${\displaystyle \langle O\vert a_{p}}$

${\displaystyle \vert \vert a_{p}^{\dagger }\vert O\rangle \vert \vert ^{2}=\langle O\vert a_{p}a_{p}^{\dagger }\vert O\rangle =\langle O\vert \left(a_{p}a_{p}^{\dagger }\vert O\rangle \right)=\langle O\vert O\rangle =1}$

The same rules of exchange apply to the annihilation operators as to the producers. Applying an annihilator to the vacuum state yields zero (the zero vector).

• The transition of a particle from the state to is accomplished by the operator . You destroy the particle in and create a new one in . This way of describing misleading questions that arise from everyday experience with macroscopic particles is avoided: ${\displaystyle p}$ ${\displaystyle k}$ ${\displaystyle a_{k}^{\dagger }a_{p}}$ ${\displaystyle p}$ ${\displaystyle k}$

1. The obvious question for everyday objects, whether the particle in the state is actually the same particle as it was before in the state , cannot be asked at all. ${\displaystyle k}$ ${\displaystyle p}$
2. The everyday question of where the particle was during the quantum leap from to cannot be asked either. ${\displaystyle p}$ ${\displaystyle k}$

• Annihilators are interchangeable with producers , unless they refer to the same state. Then: ${\displaystyle a_{p}}$ ${\displaystyle a_{k}^{\dagger }}$

${\displaystyle a_{p}a_{p}^{\dagger }=+a_{p}^{\dagger }a_{p}+1}$ for bosons ("interchangeable")

${\displaystyle a_{p}a_{p}^{\dagger }=-a_{p}^{\dagger }a_{p}+1}$ for fermions ("non-exchangeable")

• The operator that gives the number of particles present in the state as an eigenvalue is the particle number operator . It is the same for fermions and bosons. (For fermions it has no eigenvalues ​​other than 0 and 1.) ${\displaystyle p}$ ${\displaystyle {\hat {n}}_{p}=a_{p}^{\dagger }a_{p}}$

• The connection of a 1-particle state with its "old" wave function results from creating a locally localized particle (state ) and using it to form the scalar product, which indicates the amplitude of one state in the other: ${\displaystyle a_{p}^{\dagger }\vert O\rangle }$ ${\displaystyle \psi _{p}({\vec {r}})}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle a_{\vec {r}}^{\dagger }\vert O\rangle }$ ${\displaystyle a_{p}^{\dagger }\vert O\rangle }$

${\displaystyle \psi _{p}({\vec {r}})=\langle O\vert a_{\vec {r}}a_{p}^{\dagger }\vert O\rangle }$

Mathematical construction

The crucial work, configuration space and second quantization , comes from the Russian physicist Wladimir Fock in 1932.

Let be an orthonormal one-particle basis of a quantum mechanical system (i.e. a set of wave functions according to which any one-particle wave function can be expanded). Then it is known that every fermionic (or bosonic) many-body wave function, which is naturally antisymmetric (or symmetric), can be developed according to determinants (or permanent ones ) with respect to this single-particle basis ( Slater determinant ): Be antisymmetric ( , e.g. position and spin coordinates of an electron). Then there are complex numbers (i.e. for every “configuration” in which there are indices in the one-particle basis, there are complex coefficients) with ${\displaystyle \{|\phi _{j}\rangle \}_{j}}$ ${\displaystyle \Psi (x_{1},\ldots ,x_{N})}$ ${\displaystyle x_{j}=(\mathbf {r} _{j},s_{j})}$ ${\displaystyle c_{L}\in \mathbb {C} }$ ${\displaystyle L=(l_{1},\ldots ,l_{N})}$ ${\displaystyle l_{x}}$ ${\displaystyle N}$

${\displaystyle \Psi (x_{1},\ldots ,x_{N})=\sum _{L\in \mathbb {N} ^{N},{\text{ geordnet}}}c_{L}{\frac {1}{\sqrt {N}}}\det \left|\left\langle x_{j}|\phi _{l_{k}}\right\rangle \right|_{(j,k)}=\sum _{L}{\frac {c_{L}}{\sqrt {N}}}\det {\begin{pmatrix}\phi _{l_{1}}(x_{1})&\cdots &\phi _{l_{N}}(x_{1})\\\vdots &\ddots &\vdots \\\phi _{l_{1}}(x_{N})&\cdots &\phi _{l_{N}}(x_{N})\end{pmatrix}}}$

So every many-body wave function can be represented as a linear combination of such determinant states (or corresponding permanent states in the bosonic case). In addition to their purely mathematical significance as a basis for development, these determinant states are often of great physical importance, since the ground state wave functions of non-interacting systems can be represented as pure determinant states (or permanent states).

The determinant / permanent for the configuration can now be called ${\displaystyle L=(l_{1},\ldots ,l_{N})}$

${\displaystyle |0,0,\underbrace {n_{1}} _{\nwarrow l_{1}{\text{te Stelle}}},0,0,0,\underbrace {n_{2}} _{\nwarrow l_{1+n_{1}}{\text{te Stelle}}},\ldots \rangle }$

assign, with number of occurrences of the value of in , number of occurrences of the value of in ,…. The values are called the occupation numbers of the associated base states. The occupation numbers for fermions can only be 1 or 0, otherwise the determinant would disappear (two equal columns). ${\displaystyle n_{1}=}$ ${\displaystyle l_{1}}$ ${\displaystyle L}$ ${\displaystyle n_{2}=}$ ${\displaystyle l_{2}}$ ${\displaystyle L}$ ${\displaystyle n_{j}}$

In this notation, the general representation of an N-particle many-particle state is : ${\displaystyle |\Psi \rangle }$

${\displaystyle |\Psi \rangle =\sum _{n_{1},n_{2},\ldots =0;n_{1}+n_{2}+\ldots =N}^{1{\text{ bzw. }}\infty }c_{n_{1},\ldots ,n_{\infty }}|n_{1},n_{2},\ldots ,n_{\infty }\rangle }$

the occupation number representation . The antisymmetric or symmetric -particle Hilbert space is thus by these states with stretched. It is now obvious to introduce a more general space - which is called "Fock space" - which is spanned by the -states with any finite number of particles: ${\displaystyle N}$ ${\displaystyle {\mathcal {H}}_{N}}$ ${\displaystyle |n_{1},n_{2},\ldots \rangle }$ ${\displaystyle \sum n_{j}=N}$ ${\displaystyle |n_{1},n_{2},\ldots \rangle }$

${\displaystyle F:={\text{clin}}\{|n_{1},n_{2},\ldots \rangle ;\;\sum n_{j}\,{\text{endl.}}\}=\bigoplus _{N}{\mathcal {H}}_{N}}$.

Since operators can be represented independently of the concrete number of particles (see below), this construction makes sense. This space contains states of an indefinite number of particles ( linear combination of states of different, specific numbers of particles). Many-particle theory is normally practiced in it.

Individual determinant states which, as already said, z. B. could be special states of an interaction-free system, one can clearly state in the form when one says to which one-particle basis one refers. ${\displaystyle |\Psi \rangle =|n_{1},n_{2},\ldots \rangle }$

See also: Slater determinant

Creation, annihilation and particle number operators

→ Main article : Creation and annihilation operator

In order to no longer have to carry out the annoying (anti-) symmetrizing described above to generate jib states, the jib states are now generated from the vacuum state instead. To this end, new operators are introduced that “create” or “destroy” particles in the base state , i.e. That is, increase or decrease the corresponding occupation number, whereby the whole symmetry problem is now in the commutator relations: ${\displaystyle j\equiv |\phi _{j}\rangle }$

Definition (on the basis of the state space, on the rest by linear continuation):

• In the boson case

${\displaystyle c_{j}^{\dagger }:H_{N}^{S}\rightarrow H_{N+1}^{S},\quad c_{j}^{\dagger }|\ldots n_{j}\ldots \rangle :={\sqrt {n_{j}+1}}|\ldots n_{j}+1\ldots \rangle }$

${\displaystyle c_{j}:H_{N}^{S}\rightarrow H_{N-1}^{S},\quad c_{j}|\ldots n_{j}\ldots \rangle :={\sqrt {n_{j}}}|\ldots n_{j}-1\ldots \rangle }$

• In the fermionic case

${\displaystyle c_{j}^{\dagger }:H_{N}^{A}\rightarrow H_{N+1}^{A},\quad c_{j}^{\dagger }|\ldots n_{j}\ldots \rangle :=(-1)^{\sum _{i<j}n_{i}}\;(1-n_{j})|\ldots \underbrace {n_{j}+1} _{=1}\ldots \rangle }$

${\displaystyle c_{j}:H_{N}^{A}\rightarrow H_{N-1}^{A},\quad c_{j}|\ldots n_{j}\ldots \rangle :=(-1)^{\sum _{i<j}n_{i}}\;n_{j}|\ldots \underbrace {n_{j}-1} _{=0}\ldots \rangle }$

The pre-factors ensure that impossible states do not occur (e.g. with occupation numbers less than 0 or greater than 1 for fermions), for the encapsulation of the antisymmetry for fermions in other expressions and for the fact that the occupation number operators in both cases are

${\displaystyle {\hat {n}}_{j}:=c_{j}^{\dagger }c_{j}}$

surrender. Recalculation shows that these operators reproduce the occupation numbers for determinant states:

${\displaystyle {\hat {n}}_{j}|\ldots ,n_{j},\ldots \rangle =n_{j}|\ldots ,n_{j},\ldots \rangle }$.

Commutation relations

For the operators constructed in this way, the anti-commutation relations apply in the fermionic case

${\displaystyle \{c_{i},c_{j}^{\dagger }\}=\delta _{ij}\qquad \{c_{i},c_{j}\}=0\qquad \{c_{i}^{\dagger },c_{j}^{\dagger }\}=0,}$

where denotes the anti- commutator. ${\displaystyle \{A,B\}:=AB+BA}$

In the boson case, the exchange relations apply

${\displaystyle [c_{i},c_{j}^{\dagger }]=\delta _{ij}\qquad [c_{i},c_{j}]=0\qquad [c_{i}^{\dagger },c_{j}^{\dagger }]=0.}$

Inside is the commutator . ${\displaystyle [A,B]:=AB-BA}$

One- and two-particle operators

It can be shown that all linear operators on the Fock space can be represented as linear combinations of polynomials in the creation / annihilation operators. The so-called one-particle or two-particle operators are important here, which according to their name represent either observables of individual particles (e.g. kinetic energy, position, spin) or interactions between two particles (e.g. Coulomb interaction between two electrons ).

This results in simple expressions ( is the number of particles): Let ${\displaystyle N}$

${\displaystyle A=\sum _{\alpha =1}^{N}h_{\alpha }\,}$

a one-particle operator (i.e. each acts only on the coordinates of the -th particle, but in terms of structure they are all the same), then (by inserting ones and using the relation valid for bosons and fermions , where the index is a Characterizes the single-particle state in the Hilbert space of the -th particle): ${\displaystyle h_{\alpha }\,}$ ${\displaystyle \alpha \,}$ ${\displaystyle h_{\alpha }\,}$ ${\displaystyle c_{i}^{\dagger }c_{j}=\sum _{\alpha =1}^{N}|i\rangle _{\alpha }\langle j|_{\alpha }}$ ${\displaystyle |i\rangle _{\alpha }}$ ${\displaystyle i}$ ${\displaystyle \alpha }$

${\displaystyle A=\sum _{\alpha }h_{\alpha }=\sum _{i,j}\langle i|h|j\rangle c_{i}^{\dagger }c_{j}=\sum _{i,j}\langle \phi _{i}|h|\phi _{j}\rangle c_{i}^{\dagger }c_{j}}$

where the matrix element is the one-particle operator from which they result, formed with the base states with respect to which was quantized. For two-particle operators it results analogously: ${\displaystyle \langle i|h|j\rangle }$ ${\displaystyle h_{i}\,}$ ${\displaystyle |\phi _{j}\rangle }$

${\displaystyle A=\sum _{\alpha <\beta }w(\alpha ,\beta )={\frac {1}{2}}\sum _{\alpha ,\beta \neq \alpha }w(\alpha ,\beta )={\frac {1}{2}}\sum _{i,j,k,l}\langle ij|w|lk\rangle c_{i}^{\dagger }c_{j}^{\dagger }c_{k}c_{l}}$ ${\displaystyle ={\frac {1}{2}}\sum _{i,j,k,l}\langle \phi _{i}^{(1)}\phi _{j}^{(2)}|w(1,2)|\phi _{l}^{(1)}\phi _{k}^{(2)}\rangle c_{i}^{\dagger }c_{j}^{\dagger }c_{k}c_{l}}$.

The expressions are true equality of operators as long as they are related to a fixed number of particles. But you can see that the second quantized form of the operators no longer explicitly contains the particle number. The second quantized operators take on the same form in systems with different numbers of particles.

Concrete examples

One-particle operators

Particle density in second quantization with respect to momentum basis (discrete momentum basis, finite volume with periodic boundary conditions ):

${\displaystyle \rho (r)=\sum _{\alpha =1}^{N}\delta (r-{\hat {x}}_{\alpha })}$

${\displaystyle =\sum _{k,k'}\langle k|\delta (r-{\hat {x}})|k'\rangle c_{k}^{\dagger }c_{k'}}$

${\displaystyle =\sum _{k,k'}\int _{x\in V}\mathrm {d} ^{3}x\,\langle k|\delta (r-{\hat {x}})|x\rangle \langle x|\,|k'\rangle c_{k}^{\dagger }c_{k'}}$

${\displaystyle =\sum _{k,k'}\int _{x\in V}\mathrm {d} ^{3}x\,\langle k|x\rangle \delta (r-x)\langle x|k'\rangle c_{k}^{\dagger }c_{k'}}$

${\displaystyle =\sum _{k,k'}\int _{x\in V}\mathrm {d} ^{3}x\,{\frac {1}{V}}e^{i(k'-k)x}\delta (r-x)c_{k}^{\dagger }c_{k'}}$

${\displaystyle =\sum _{k,k'}{\frac {1}{V}}e^{i(k'-k)r}c_{k}^{\dagger }c_{k'}}$

${\displaystyle ={\frac {1}{V}}\sum _{k,q}e^{iqr}c_{k}^{\dagger }c_{k+q}}$

Coulomb interaction

In second quantization with respect to (discrete) impulse basis.

${\displaystyle W_{\text{Coul.}}={\frac {1}{2}}\sum _{\alpha ,\alpha \neq \beta }{\frac {e^{2}}{|\mathbf {r_{\alpha }-r_{\beta }|} }}={\frac {1}{2V}}\sum _{q\neq 0,k_{1},\sigma _{1},k_{2},\sigma _{2}}{\frac {4\pi e^{2}}{q^{2}}}c_{k_{1},\sigma _{1}}^{\dagger }c_{k_{2},\sigma _{2}}^{\dagger }c_{k_{2}-q,\sigma _{2}}c_{k_{1}+q,\sigma _{1}}}$

Superconductivity

With the Fock representation, the second quantization also enables states to be explicitly taken into account that are not   eigen-states of the particle number operator. Such states play a major role in the theory of superconductivity . ${\displaystyle {\hat {N}}=\sum _{k,\sigma }c_{k,\sigma }^{\dagger }c_{k,\sigma }}$

Transformation between single-particle bases

Creation and annihilation operators with respect to a given one-particle basis can be expressed by corresponding operators with respect to another one-particle basis : ${\displaystyle |i\rangle }$ ${\displaystyle |\alpha \rangle }$

${\displaystyle c_{i}^{\dagger }=\sum _{\alpha }\langle \alpha |i\rangle c_{\alpha }^{\dagger }}$

${\displaystyle c_{i}=\sum _{\alpha }\langle i|\alpha \rangle c_{\alpha }.}$

Through these relationships, it is possible to change the base in the Fock space and thus transform given expressions to forms that are more suitable for the current situation. In a similar way, field operators with respect to continuous position or momentum bases are generated from the creation / annihilation operators for discrete one-particle bases, as they are mainly used in quantum field theories .

Generalization: Relativistic quantum field theories

As a generalization, as indicated in the footnote, relativistic quantum field theories emerge instead of the non-relativistic many-body theory .

literature

• Alexander Altland, Ben Simons: Condensed matter field theory , Cambridge Univ. Press, 2009, ISBN 978-0-521-84508-3
• Eugen Fick: Introduction to the Basics of Quantum Theory , Wiesbaden, 1988, ISBN 3-89104-472-0
• Wolfgang Nolting: Basic course in theoretical physics , Volume 7: Many-particle physics , Berlin a. a., 2009, ISBN 978-3-642-01605-9
• Franz Schwabl: Quantum Mechanics for Advanced Students (QM II) , Berlin a. a., 2008, ISBN 978-3-540-85075-5

References and footnotes

1. ↑ PAM Dirac, Proc. R. Soc. London A 114 , 243-265 (1927) doi : 10.1098 / rspa.1927.0039
2. ^ W. Fock, Zeitschrift für Physik 75 , 622– 647 (1932) doi : 10.1007 / BF01344458
3. ↑ W. Nolting, Basic Course Theoretical Physics 6 (Springer, 2014), p. 177, doi : 10.1007 / 978-3-642-25393-5
4. ↑ The second quantization can also be formulated as a field quantization of a certain classical field, which is compatible with the Schrödinger equation, the so-called "Schrödinger field". Instead of the Schrödinger equation, one can also deal with relativistic classical equations compatible with quantum theory or their field theories. The resulting equations would be e.g. B. in the structure analogous to those of Maxwell's theory and must in the special cases of the Schrödinger field or the so-called. QED or QCD u. a. contain Maxwell's field energy as a contribution to the potential energy of the electrons, but also Planck's constant h as a field parameter in their kinetic energy . Instead of the non-relativistic many-body theory, relativistic quantum field theories emerge .